n阶非驻定系统的渐近稳定性

The Asymptotic Stability of Nonstationary Systeme

用第二方法判定非驻定系统号ax/(dt)=f(t、x)(其中x、f均为n维向量函数)零解的渐近稳定性时,需要构造一个具有无穷小上界的正定函数且要满足全导数dv/(dt)<0,这无疑是困难的。本文应用[1]中的定理一,从另一个角度出发,即如果能够找到一个包围着坐标原点的运动的闭曲面Xc_0(t)(以时间t为参数),当t→∞时,此闭曲面收缩到原点,而且在t_0时从闭曲面Xc_0(t)内原点附近发出的轨道不能与闭曲面Xc_0(t)相交,那么随着运动的闭曲面的收缩,将迫使轨道趋于原点。因而就可以断定零解是渐近稳定的。在此基础上给出一类n阶非驻定系统的渐近稳定性的判定准则。

In order to determine by Lyapunov's second method the asymptotic stability of zero solution of nonstationary system (dx)/(dt)=f(t, x)where both of x and f are n-dimensional vector functions, we have to construct a positive definite function with infinitesimally small upper bound, satisfying total derivative (dv)/(dt)= 0. It is undoubtedly difficult to do so.This paper solves the problem from another point of view, using Th. 1 of (l).If we can find a closed surface Xco(t) with time t a parameter , moving around origin and contracting to origin and the trajectory starting at time t_0 near by origin inside the closed surface Xco(t) could not intersect the closed surface Xco(t), then the trajectory, accompanying the contraction of the moving surface, will be compelled to tend to origin. Thus we can assert that the zero solution is stable.On this basis we give a criterion of asymptotic stability of a class of nonstationary system with order n.